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166       XIII    INTEGRATION

8. (a) Let (/*)i^^« be a finite orthonormal sequence in ^c(X, /*)• Show that there
exists a constant c independent of n such that, for every finite sequence (tf*)io<n of
complex numbers, we have

sup

j

(Suppose first that n — 2r. For each 7 g /?, let .s/ = Z a*/>" Split UP tn*s sum mto at
most r + 1 partial sums corresponding to the dyadic expansion of the number ;, so

r

that Sj = 2 pq where, for each q, either pq = 0 or else pq is of the form

q = Q

s    «*/».

with 0 <: h < 2r~*. Using Cauchy's inequality |^|2 <^ r • X l^l2, show that

«=o

Finally, if 2r <; n< 2r+1, take the <2k to be 0 for k > «.)

(b) Let (w(n))nzi be an increasing sequence of integers >0, and tending to +cx>.
Let (/„)«£! be an orthonormal sequence in ^c(^j M)- If *ne sequence («„)„>!
is such that X W(w)l^nl2< +°°> and if (wfc) is a sequence of integers such that

k <^ w(rik) <k+l9 then the sequence of partial sums snk = J] «,/// converges almost

everywhere in X. (If /e ^c(X, ju-) has the fln as coefficients (6.5.2) with respect to the
system (/„), evaluate N2(/— snk) — rnk say, and remark that the series with general
term r Jk converges, using Abel's partial summation technique by writing rjfc =
(k + 1— k)r*k (Section 13.21, Problem 6).)

(c)   Suppose that the sequence (an) is such that]£ |^n|2(log n)2 < +00. Show that the

n

series ^ anfn converges almost everywhere ("Rademacher-MenchofT theorem")-

n

(First deduce from (b) that the sequence (s2k) converges almost everywhere. Then
use (a) to show that, if Sfc = sup \sn — s2k\, the series with general term Si du,

2fc4«<2*+1                                                                         J

converges.)

9. Suppose that the measure /x is bounded, so that L2(X, /A) <= LJ(X, p). Suppose also
that jLi(X)= 1. Let U be a continuous endomorphism of L1 with norm ^1 (a "con-
traction ") whose restriction to L2 is also a contraction (i.e., N2(t/ •/) ^ N2(7) in L2).

1 n~1

Let P be the endomorphism of L2 which is the strong limit of - T Uk (Section 12.1 5,

n *=o

Problem 12), so that P = PU = UP =P2. Show that P can be extended by con-
tinuity to;an endomorphism of L1 satisfying the same relations and that

P-/«lim -ZV-/

n -»• oo n k =s o

for all/e L1 (use the fact that L2 is dense in L1 and that N^/) ^ N2(/).) with general  term  anfn(x)